Mathematical analysis, also known as advanced calculus, is the oldest and most basic branch of analysis. Generally speaking, it refers to a relatively complete mathematical subject with the general theory of calculus and infinite series as the main content, including their theoretical basis (basic theory of real number, function and limit).

It is also a basic course for college mathematics majors. The branch of analysis in mathematics is a branch of mathematics that specializes in studying real numbers and complex numbers and their functions.

2. Advanced Algebra

Elementary algebra begins with the simplest one-dimensional linear equation. On the one hand, elementary algebra further discusses binary and ternary linear equations, on the other hand, it studies equations that are larger than quadratic and can be reduced to quadratic. Along these two directions, algebra discussed any number of unknown linear equations, also known as linear equations, and also studied higher-order unary equations.

This stage is called advanced algebra. Advanced algebra is a general term for the development of algebra to an advanced stage, including many branches. Higher algebra offered by universities now generally includes two parts: linear algebra and polynomial algebra.

3. Analytic geometry

Analytic geometry was founded and developed by mathematicians such as Descartes and Fermat with the help of Cartesian coordinate system. It is a branch of geometry, which uses analytical expressions to study the relations and properties between geometric objects, also known as coordinate geometry.

Strictly speaking, analytic geometry does not use algebraic methods, but uses analytical formulas to study geometric figures. Analytic expressions can be algebraic or transcendental, such as trigonometric functions, logarithms, etc. Usually, the default algebraic expression only consists of four operations and the roots of finite steps, and transcendental operation generally does not belong to the research field of algebra.

4. Abstract algebra

Abstract algebra (abstract algebra)

Also known as modern algebra, it came into being in19th century. Galois [181-1832] used the concept of "group" in1832, which completely solved the possibility of solving rooted algebraic equations.

He was the first mathematician who put forward the concept of "group" and is generally called the founder of modern algebra. He changed algebra from the science of solving equations to the science of studying algebraic operation structure, that is, he pushed algebra from elementary algebra to abstract algebra.

5. Theory of real variable function

The theory of real variable function is a branch of mathematics formed at the end of 19 and the beginning of the 20th century. Originated from classical analysis, the main research object is the function of independent variables (including multivariate) taking real values. The problems studied include the basic theories of function continuity, differentiability, integrability and convergence, which are the deepening and development of calculus.

Because it studies not only the functions in calculus, but also more general functions, and draws deeper and more general conclusions than the corresponding theories in calculus, the theory of real variable functions is the basis of all branches of modern analytical mathematics.